Equation (REF ) is coupled to a rate equation describing
the evolution of the number of excited molecules due to emission and
absorption processes and external pumping. The latter has to be
applied to compensate for the photon losses. The steady-state
average value \(\bar{n}\)  of the photon number in each cavity, \(n =|\psi (\mathbf {x})|^2\) , results from the balance between the pumping rate
\(P\)  and losses and satisfies \(P = \gamma \bar{n}\) .  Under the
condition \(J \ll T\) , which assures that the occupations of all
momentum states are much larger than one, the generalized Gross
Pitaevskii classical field model (REF ) is valid for all the
modes and there is no need to use a more refined quantum optical
approach [1]}, [2]}, [3]}.
